Quantum Plücker Embeddings

• Quantum Plücker embeddings are algebraic and geometric structures that generalize classical Plücker embeddings to quantum, deformed, and operator-theoretic contexts.
• They integrate methods from algebraic geometry, representation theory, and integrable systems to model quantum cohomology rings and mirror symmetry through deformed Plücker relations.
• Their applications span quantum information theory, quantum many-body systems, and operator algebras, offering new insights into quantum moduli spaces and integrability.

Quantum Plücker embeddings are algebraic, geometric, and representation-theoretic structures that generalize the classical Plücker embedding of Grassmannians and flag varieties to quantum, deformed, or operator-theoretic settings. These constructions play a central role in modern mathematical physics, algebraic geometry, and quantum information theory, underpinning deep connections between quantum cohomology, mirror symmetry, integrable systems, and quantum groups.

1. Classical Plücker Embeddings and Their Extensions

The classical Plücker embedding realizes the Grassmannian $\mathrm{Gr}(k,n)$ as a projective algebraic variety in $\mathbb{P}(\wedge^k \mathbb{C}^n)$ using the wedge products of basis vectors. The quadratic Plücker relations among minors cut out the Grassmannian as a subvariety. In the context of flag varieties, sequences of subspaces are embedded into products of projective spaces via Plücker coordinates, with their ideal of relations governing the structure sheaf of the embedded variety.

Quantum Plücker embeddings fundamentally extend this setup. They relate to:

• The coordinate rings of quantum Grassmannians and quantum flag varieties (deformations of the classical varieties).
• Structures encoding the quantum cohomology ring and its B-model mirror dual.
• Deformations arising in the theory of quantum groups.
• Operator-valued analogues relevant for quantum integrable systems and quantum information theory.

The necessity for such quantum versions arises in several contexts:

• Mirror symmetry for flag varieties, where the mirror Landau-Ginzburg model is constructed via superpotentials expressed in Plücker coordinates, and the Jacobi ring (the ring of functions modulo critical equations) models quantum cohomology (Li et al., 28 Jan 2024, Kalashnikov, 2020).
• Representation theory, where semi-infinite or quantum Plücker relations define coordinate algebras isomorphic to modules for current or quantum algebras (Feigin et al., 2017).
• Quantum many-body systems and integrability, where commuting operator-valued matrices obey Plücker-type constraints (Shastry, 2010).

2. Quantum Plücker Embeddings in Mirror Symmetry and Quantum Cohomology

For type A partial flag varieties, explicit quantum Plücker superpotentials $\mathcal{F}_-$ are constructed in Plücker coordinates, giving: $$\mathcal{F}_-(P, q) = \sum_{i\in I_P} q_i v_{i,i+1}(P) + \sum_{i=1}^{n-1} u_{i,i+1}(P)$$ where $v_{i,i+1}$ and $u_{i,i+1}$ are rational functions (quotients) of Plücker coordinates, and $q_i$ are quantum parameters determined by the sequence of the flag. The domain is an open subset of the flag variety defined by the nonvanishing of specific Plücker combinations, concretely encoding the anti-canonical divisor structure (Li et al., 28 Jan 2024).

The Jacobi ring associated with $\mathcal{F}_-$,

$$\operatorname{Jac}(\mathcal{F}_-) = \mathcal{O}((P \backslash G)^\circ)/ \left( \frac{\partial \mathcal{F}_-}{\partial z_j} \right),$$

canonically models the small quantum cohomology ring $QH^*(\mathbb{F}\ell(n_\bullet))$. Quantum Plücker embeddings manifest here as the canonical map from Schubert classes in quantum cohomology to Plücker coordinate functions in this Jacobi ring, potentially involving quantum corrections (additional terms or deformed products).

The critical values of $\mathcal{F}_-$ match the spectrum of quantum multiplication by the first Chern class $c_1$, confirming a longstanding conjecture in mirror symmetry and revealing the deep algebraic structure encoded by quantum Plücker coordinates (Li et al., 28 Jan 2024).

3. Quantum Schubert Calculus and Skew Plücker Relations

Quantum Schubert calculus elucidates the relations among quantum Schubert classes, which, upon identification with Plücker coordinates, become "quantum Plücker" relations. These relations are not simply quadratic in the original variables but are deformed by quantum parameters and feature higher combinatorial complexity, particularly for partial flag varieties.

Recent advances generalize classical quadratic Plücker relations to "skew Plücker relations," involving products and sums of skew Schur functions. These encode relations in the coordinate rings of more general or infinite-dimensional flag varieties and are reflected in the quadratic (and higher) quantum relations among Plücker-type coordinates in the quantum setting (Aokage et al., 11 Apr 2025). This framework is closely linked with Hirota bilinear equations and integrable systems, as both the classical and skew Plücker relations correspond to bilinear constraints on tau functions in the KP and mKP hierarchies.

A summary of the extensions is given in the following table:

Structure	Schur/Plücker Relations	Skew/Quantum Generalization	Quantum Context
Flag embedding	Minors/Schur, quadratic	Skew Schur (quadratic/higher), relatives	Quantum minors, quantum Schur functions
Integrable systems	KP Hirota equations	mKP/higher Hirota/skew Hirota	Quantum tau-functions
Cohomology ring	Classical cup product	Quantum (deformed) product	$QH^*(\mathbb{F}\ell)$, Jacobi ring

4. Quantum Plücker Embeddings in Representation Theory

In representation theory, quantum Plücker embeddings are realized as identifications between coordinate rings of (possibly infinite-dimensional) flag varieties or quotients by Plücker-type ideals and the sum of dual (global) Weyl modules. For the semi-infinite setting, the homogeneous coordinate ring $W$ is identified as

$$W \cong \bigoplus_{\lambda \in P_+} \mathbb{W}(\lambda)^*,$$

where $\mathbb{W}(\lambda)$ is the global Weyl module for the current algebra $\mathfrak{g}[t]$ (Feigin et al., 2017). The arced (or "semi-infinite") Plücker relations governing the ideal involve higher order derivatives and encodings of the infinite flag variety structure.

Quantum analogues of these structures are crucial for defining the coordinate rings and K-theoretic invariants of quantum flag varieties and relate directly to $q$-Whittaker functions, zastava spaces, and the geometry of representations for quantum groups.

5. Quantum Plücker Embeddings in Quantum Integrable Systems

Type-I matrices, introduced as finite-dimensional prototypes of quantum integrable systems, provide a concrete realization of quantum Plücker embeddings in the setting of operator algebras: $$Z(r) = |r \rangle \langle r| + x \sum_{s} [\rho_s(r) |s\rangle \langle s| + S_{r,s} (|r\rangle \langle s| + |s\rangle \langle r|)]$$ with commutativity $\forall x$ enforced precisely by the vanishing of the Plücker relation

$R_{ij} R_{kl} - R_{ik} R_{jl} + R_{il} R_{jk} = 0,$

where $R_{ij}=1/S_{ij}$ (Shastry, 2010). Upon second quantization, these matrices become operator-valued and can be realized in both Fermi and Bose settings, generating integrable quantum glass models with extensive conserved quantities.

This embedding realizes the classical Plücker (Grassmannian) geometry as exact commutation relations in the quantum theory, with the explicit parametric solution providing a full classification of integrable quantum systems in this class.

6. Constraints, Degeneracies, and Geometric Structures

Quantum Plücker embeddings are subject to further geometric and algebraic constraints:

• The stratification of Hilbert schemes via Grothendieck–Plücker embedding reveals degeneracy: for nonconstant Hilbert polynomials and high embedding degree, the image is contained in a proper subvariety of the ambient Grassmannian, a result with significant implications for quantum analogues and the search for fully faithful embeddings (Hyeon et al., 2017).
• The bilinear and higher relations in the isotropic and symplectic Grassmannian contexts (e.g., Pfaffian formulas and Cartan embeddings) generalize to quantum settings, suggesting further connections between determinants, Pfaffians, and quantum minors (Balogh et al., 2020).

7. Outlook and Open Directions

Quantum Plücker embeddings unify diverse strands in mathematical physics and algebraic geometry:

• In mirror symmetry, they furnish explicit Landau-Ginzburg models and explicit coordinate representations for quantum cohomology rings, enabling direct computation of quantum invariants (Li et al., 28 Jan 2024, Kalashnikov, 2020).
• In integrable systems and operator theory, their algebraic underpinnings provide structural explanations for integrability and localization phenomena in quantum glass and related models (Shastry, 2010).
• In combinatorics and representation theory, skew and semi-infinite Plücker structures, new character formulas, and module identifications deepen the understanding of quantum flag varieties and their module categories (Aokage et al., 11 Apr 2025, Feigin et al., 2017).
• The general framework guides the extension of classical algebraic geometry into quantum, noncommutative, and infinite-dimensional settings, drawing on new algebraic structures, deformation theory, and combinatorial generalizations.

A plausible implication is that advances in explicit quantum Plücker relations and their realization in coordinate rings and operator algebras will provide new tools for studying the geometry of quantum moduli spaces, the structure of quantum invariants, and the representation theory of quantum groups, as well as supporting the development of new quantum algorithms in information theory and mathematical physics.